Mean Convergence Theorems and Weak Laws of Large Numbers for Arrays of Measurable Operators under Some Conditions of Uniform Integrability

Abstract

In this paper, we introduce the notions of uniform integrability in the Cesàro sense, h-integrability with respect to the array of constants {a_ni}, and h-integrability with exponent r for an array of measurable operators. Then, we establish some mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions related to these notions.